# General Ring Theory/Ring extensions

**Proposition (commutative ring extension is a module)**:

Let be a commutative ring and let be a commutative extension ring of . Then with its own addition and the restriction of the multiplication of to is a module over .

**Proof:** From the axioms holding for rings, we deduce the module axioms as follows as follows: Let and . Then

- (distributivity)
- (distributivity)
- (commutativity of multiplication)
- (unit),

the ring axiom that's being used being indicated in the brackets.

**Proposition ()**:

Let be a ring extension. Then the function

defines a function from ideals of to ideals of .

**Proof:** Indeed, because it is certainly closed under addition and multiplication by elements of .

**Proposition ()**:

Let be a ring extension, and suppose that is a multiplicative set. Then is a multiplicative set of .

**Proof:** is closed under multiplication because both and are.